add Constructions module with Direct/Wreath product

Project.toml

@@ -8,6 +8,7 @@ Folds = "41a02a25-b8f0-4f67-bc48-60067656b558"
 GroupsCore = "d5909c97-4eac-4ecc-a3dc-fdd0858a4120"
 KnuthBendix = "c2604015-7b3d-4a30-8a26-9074551ec60a"
 OrderedCollections = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
+PermutationGroups = "8bc5a954-2dfc-11e9-10e6-cd969bffa420"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 
 [compat]

src/Groups.jl

@@ -15,6 +15,10 @@ export Alphabet, AutomorphismGroup, FreeGroup, FreeGroup, FPGroup, FPGroupElemen
 
 export alphabet, evaluate, word, gens
 
+# general constructions
+include(joinpath("constructions", "constructions.jl"))
+using .Constructions
+
 include("types.jl")
 include("hashing.jl")
 include("normalform.jl")

src/constructions/constructions.jl

@@ -0,0 +1,10 @@
+module Constructions
+
+using GroupsCore
+using Random
+
+include("direct_product.jl")
+include("direct_power.jl")
+include("wreath_product.jl")
+
+end # of module Constructions

src/constructions/direct_power.jl

@@ -0,0 +1,112 @@
+struct DirectPower{Gr,N,GEl<:GroupsCore.GroupElement} <: GroupsCore.Group
+    group::Gr
+
+    function DirectPower{N}(G::GroupsCore.Group) where {N}
+        @assert N > 1
+        return new{typeof(G),N,eltype(G)}(G)
+    end
+end
+
+struct DirectPowerElement{GEl,N,Gr<:GroupsCore.Group} <: GroupsCore.GroupElement
+    elts::NTuple{N,GEl}
+    parent::DirectPower{Gr,N,GEl}
+end
+
+DirectPowerElement(
+    elts::AbstractVector{<:GroupsCore.GroupElement},
+    G::DirectPower,
+) = DirectPowerElement(ntuple(i -> elts[i], _nfold(G)), G)
+
+_nfold(::DirectPower{Gr,N}) where {Gr,N} = N
+
+Base.one(G::DirectPower) =
+    DirectPowerElement(ntuple(_ -> one(G.group), _nfold(G)), G)
+
+Base.eltype(::Type{<:DirectPower{Gr,N,GEl}}) where {Gr,N,GEl} =
+    DirectPowerElement{GEl,N,Gr}
+
+function Base.iterate(G::DirectPower)
+    itr = Iterators.ProductIterator(ntuple(i -> G.group, _nfold(G)))
+    res = iterate(itr)
+    @assert res !== nothing
+    elt = DirectPowerElement(first(res), G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.iterate(G::DirectPower, state)
+    itr, st = state.iterator, state.state
+    res = iterate(itr, st)
+    res === nothing && return nothing
+    elt = DirectPowerElement(first(res), G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.IteratorSize(::Type{<:DirectPower{Gr,N}}) where {Gr,N}
+    Base.IteratorSize(Gr) isa Base.HasLength && return Base.HasShape{N}()
+    Base.IteratorSize(Gr) isa Base.HasShape && return Base.HasShape{N}()
+    return Base.IteratorSize(Gr)
+end
+
+Base.size(G::DirectPower) = ntuple(_ -> length(G.group), _nfold(G))
+
+GroupsCore.order(::Type{I}, G::DirectPower) where {I<:Integer} =
+    convert(I, order(I, G.group)^_nfold(G))
+
+GroupsCore.ngens(G::DirectPower) = _nfold(G)*ngens(G.group)
+
+function GroupsCore.gens(G::DirectPower, i::Integer)
+    k = ngens(G.group)
+    ci = CartesianIndices((k, _nfold(G)))
+    @boundscheck checkbounds(ci, i)
+    r, c = Tuple(ci[i])
+    tup = ntuple(j -> j == c ? gens(G.group, r) : one(G.group), _nfold(G))
+    return DirectPowerElement(tup, G)
+end
+
+function GroupsCore.gens(G::DirectPower)
+    N = _nfold(G)
+    S = gens(G.group)
+    tups = [ntuple(j->(i == j ? s : one(s)), N) for i in 1:N for s in S]
+
+    return [DirectPowerElement(elts, G) for elts in tups]
+end
+
+Base.isfinite(G::DirectPower) = isfinite(G.group)
+
+function Base.rand(
+    rng::Random.AbstractRNG,
+    rs::Random.SamplerTrivial{<:DirectPower},
+)
+    G = rs[]
+    return DirectPowerElement(rand(rng, G.group, _nfold(G)), G)
+end
+
+GroupsCore.parent(g::DirectPowerElement) = g.parent
+
+Base.:(==)(g::DirectPowerElement, h::DirectPowerElement) =
+    (parent(g) === parent(h) && g.elts == h.elts)
+
+Base.hash(g::DirectPowerElement, h::UInt) = hash(g.elts, hash(parent(g), h))
+
+Base.deepcopy_internal(g::DirectPowerElement, stackdict::IdDict) =
+    DirectPowerElement(Base.deepcopy_internal(g.elts, stackdict), parent(g))
+
+Base.inv(g::DirectPowerElement) = DirectPowerElement(inv.(g.elts), parent(g))
+
+function Base.:(*)(g::DirectPowerElement, h::DirectPowerElement)
+    @assert parent(g) === parent(h)
+    return DirectPowerElement(g.elts .* h.elts, parent(g))
+end
+
+GroupsCore.order(::Type{I}, g::DirectPowerElement) where {I<:Integer} =
+    convert(I, reduce(lcm, (order(I, h) for h in g.elts), init = one(I)))
+
+Base.isone(g::DirectPowerElement) = all(isone, g.elts)
+
+function Base.show(io::IO, G::DirectPower)
+    n = _nfold(G)
+    nn = n == 1 ? "1-st" : n == 2 ? "2-nd" : n == 3 ? "3-rd" : "$n-th"
+    print(io, "Direct $(nn) power of $(G.group)")
+end
+Base.show(io::IO, g::DirectPowerElement) =
+    print(io, "( ", join(g.elts, ", "), " )")

src/constructions/direct_product.jl

@@ -0,0 +1,105 @@
+using Random
+using GroupsCore
+
+struct DirectProduct{Gt,Ht,GEl,HEl} <: GroupsCore.Group
+    first::Gt
+    last::Ht
+
+    function DirectProduct(G::GroupsCore.Group, H::GroupsCore.Group)
+        return new{typeof(G),typeof(H),eltype(G),eltype(H)}(G, H)
+    end
+end
+
+struct DirectProductElement{GEl,HEl,Gt,Ht} <: GroupsCore.GroupElement
+    elts::Tuple{GEl,HEl}
+    parent::DirectProduct{Gt,Ht,GEl,HEl}
+end
+
+DirectProductElement(g, h, G::DirectProduct) = DirectProduct((g, h), G)
+
+Base.one(G::DirectProduct) =
+    DirectProductElement((one(G.first), one(G.last)), G)
+
+Base.eltype(::Type{<:DirectProduct{Gt,Ht,GEl,HEl}}) where {Gt,Ht,GEl,HEl} =
+    DirectProductElement{GEl,HEl,Gt,Ht}
+
+function Base.iterate(G::DirectProduct)
+    itr = Iterators.product(G.first, G.last)
+    res = iterate(itr)
+    @assert res !== nothing
+    elt = DirectProductElement(first(res), G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.iterate(G::DirectProduct, state)
+    itr, st = state.iterator, state.state
+    res = iterate(itr, st)
+    res === nothing && return nothing
+    elt = DirectProductElement(first(res), G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.IteratorSize(::Type{<:DirectProduct{Gt,Ht}}) where {Gt,Ht}
+    Gi = Base.IteratorSize(Gt)
+    Hi = Base.IteratorSize(Ht)
+    if Gi isa Base.IsInfinite || Hi isa Base.IsInfinite
+        return Base.IsInfinite()
+    elseif Gi isa Base.SizeUnknown || Hi isa Base.SizeUnknown
+        return Base.SizeUnknown()
+    else
+        return Base.HasShape{2}()
+    end
+end
+
+Base.size(G::DirectProduct) = (length(G.first), length(G.last))
+
+GroupsCore.order(::Type{I}, G::DirectProduct) where {I<:Integer} =
+    convert(I, order(I, G.first) * order(I, G.last))
+
+GroupsCore.ngens(G::DirectProduct) = ngens(G.first) + ngens(G.last)
+
+function GroupsCore.gens(G::DirectProduct)
+    gens_first = [DirectProductElement((g, one(G.last)), G) for g in gens(G.first)]
+
+    gens_last = [DirectProductElement((one(G.first), g), G) for g in gens(G.last)]
+
+    return [gens_first; gens_last]
+end
+
+Base.isfinite(G::DirectProduct) = isfinite(G.first) && isfinite(G.last)
+
+function Base.rand(
+    rng::Random.AbstractRNG,
+    rs::Random.SamplerTrivial{<:DirectProduct},
+)
+    G = rs[]
+    return DirectProductElement((rand(rng, G.first), rand(rng, G.last)), G)
+end
+
+GroupsCore.parent(g::DirectProductElement) = g.parent
+
+Base.:(==)(g::DirectProductElement, h::DirectProductElement) =
+    (parent(g) === parent(h) && g.elts == h.elts)
+
+Base.hash(g::DirectProductElement, h::UInt) = hash(g.elts, hash(parent(g), h))
+
+Base.deepcopy_internal(g::DirectProductElement, stackdict::IdDict) =
+    DirectProductElement(Base.deepcopy_internal(g.elts, stackdict), parent(g))
+
+Base.inv(g::DirectProductElement) =
+    DirectProductElement(inv.(g.elts), parent(g))
+
+function Base.:(*)(g::DirectProductElement, h::DirectProductElement)
+    @assert parent(g) === parent(h)
+    return DirectProductElement(g.elts .* h.elts, parent(g))
+end
+
+GroupsCore.order(::Type{I}, g::DirectProductElement) where {I<:Integer} =
+    convert(I, lcm(order(I, first(g.elts)), order(I, last(g.elts))))
+
+Base.isone(g::DirectProductElement) = all(isone, g.elts)
+
+Base.show(io::IO, G::DirectProduct) =
+    print(io, "Direct product of $(G.first) and $(G.last)")
+Base.show(io::IO, g::DirectProductElement) =
+    print(io, "( $(join(g.elts, ",")) )")

src/constructions/wreath_product.jl

@@ -0,0 +1,136 @@
+import PermutationGroups: AbstractPermutationGroup, AbstractPerm, degree, SymmetricGroup
+
+"""
+    WreathProduct(G::Group, P::AbstractPermutationGroup) <: Group
+Return the wreath product of a group `G` by permutation group `P`, usually
+written as `G ≀ P`.
+
+As set `G ≀ P` is the same as `Gᵈ × P` and the group can be understood as a
+semi-direct product of `P` acting on `d`-fold cartesian product of `G` by
+permuting coordinates. To be more precise, the multiplication inside wreath
+product is defined as
+>  `(n, σ) * (m, τ) = (n*(m^σ), σ*τ)`
+where `m^σ` denotes the action (from the right) of the permutation `σ` on
+`d`-tuples of elements from `G`.
+"""
+struct WreathProduct{DP<:DirectPower,PGr<:AbstractPermutationGroup} <:
+       GroupsCore.Group
+    N::DP
+    P::PGr
+
+    function WreathProduct(G::Group, P::AbstractPermutationGroup)
+        N = DirectPower{degree(P)}(G)
+        return new{typeof(N),typeof(P)}(N, P)
+    end
+end
+
+struct WreathProductElement{
+    DPEl<:DirectPowerElement,
+    PEl<:AbstractPerm,
+    Wr<:WreathProduct,
+} <: GroupsCore.GroupElement
+    n::DPEl
+    p::PEl
+    parent::Wr
+
+    function WreathProductElement(
+        n::DirectPowerElement,
+        p::AbstractPerm,
+        W::WreathProduct,
+    )
+        new{typeof(n),typeof(p),typeof(W)}(n, p, W)
+    end
+end
+
+Base.one(W::WreathProduct) = WreathProductElement(one(W.N), one(W.P), W)
+
+Base.eltype(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr} =
+    WreathProductElement{eltype(DP),eltype(PGr),WreathProduct{DP,PGr}}
+
+function Base.iterate(G::WreathProduct)
+    itr = Iterators.product(G.N, G.P)
+    res = iterate(itr)
+    @assert res !== nothing
+    elt = WreathProductElement(first(res)..., G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.iterate(G::WreathProduct, state)
+    itr, st = state.iterator, state.state
+    res = iterate(itr, st)
+    res === nothing && return nothing
+    elt = WreathProductElement(first(res)..., G)
+    return elt, (iterator = itr, state = last(res))
+end
+
+function Base.IteratorSize(::Type{<:WreathProduct{DP,PGr}}) where {DP,PGr}
+    dpI = Base.IteratorSize(DP)
+    pgI = Base.IteratorSize(PGr)
+
+    if dpI isa Base.IsInfinite || pgI isa Base.IsInfinite
+        return Base.IsInfinite()
+    elseif dpI isa Base.SizeUnknown || pgI isa Base.SizeUnknown
+        return Base.SizeUnknown()
+    else
+        return Base.HasShape{2}()
+    end
+end
+
+Base.size(G::WreathProduct) = (length(G.N), length(G.P))
+
+GroupsCore.order(::Type{I}, G::WreathProduct) where {I<:Integer} =
+    convert(I, order(I, G.N) * order(I, G.P))
+
+function GroupsCore.gens(G::WreathProduct)
+    N_gens = [WreathProductElement(n, one(G.P), G) for n in gens(G.N)]
+    P_gens = [WreathProductElement(one(G.N), p, G) for p in gens(G.P)]
+    return [N_gens; P_gens]
+end
+
+Base.isfinite(G::WreathProduct) = isfinite(G.N) && isfinite(G.P)
+
+function Base.rand(
+    rng::Random.AbstractRNG,
+    rs::Random.SamplerTrivial{<:WreathProduct},
+)
+
+    G = rs[]
+    return WreathProductElement(rand(rng, G.N), rand(rng, G.P), G)
+end
+
+GroupsCore.parent(g::WreathProductElement) = g.parent
+
+Base.:(==)(g::WreathProductElement, h::WreathProductElement) =
+    parent(g) === parent(h) && g.n == h.n && g.p == h.p
+
+Base.hash(g::WreathProductElement, h::UInt) =
+    hash(g.n, hash(g.p, hash(g.parent, h)))
+
+function Base.deepcopy_internal(g::WreathProductElement, stackdict::IdDict)
+    return WreathProductElement(
+        Base.deepcopy_internal(g.n, stackdict),
+        Base.deepcopy_internal(g.p, stackdict),
+        parent(g),
+    )
+end
+
+_act(p::AbstractPerm, n::DirectPowerElement) =
+    DirectPowerElement(n.elts^p, parent(n))
+
+function Base.inv(g::WreathProductElement)
+    pinv = inv(g.p)
+    return WreathProductElement(_act(pinv, inv(g.n)), pinv, parent(g))
+end
+
+function Base.:(*)(g::WreathProductElement, h::WreathProductElement)
+    @assert parent(g) === parent(h)
+    return WreathProductElement(g.n * _act(g.p, h.n), g.p * h.p, parent(g))
+end
+
+Base.isone(g::WreathProductElement) = isone(g.n) && isone(g.p)
+
+Base.show(io::IO, G::WreathProduct) =
+    print(io, "Wreath product of $(G.N.group) by $(G.P)")
+Base.show(io::IO, g::WreathProductElement) = print(io, "( $(g.n)≀$(g.p) )")
+
+Base.copy(g::WreathProductElement) = WreathProductElement(g.n, g.p, parent(g))

src/matrix_groups/MatrixGroups.jl

@@ -1,11 +1,13 @@
 module MatrixGroups
 
+using StaticArrays
+
 using GroupsCore
 using Groups
 using KnuthBendix
 
-using LinearAlgebra # Identity matrix
+import LinearAlgebra # Identity matrix
-using Random # GroupsCore rand
+import Random # GroupsCore rand
 
 export SpecialLinearGroup, SymplecticGroup
 

src/matrix_groups/abstract.jl

@@ -1,7 +1,8 @@
 abstract type MatrixGroup{N, T} <: Groups.AbstractFPGroup end
 const MatrixGroupElement{N, T} = Groups.AbstractFPGroupElement{<:MatrixGroup{N, T}}
 
-Base.isone(g::MatrixGroupElement{N, T}) where {N, T} = matrix_repr(g) == I
+Base.isone(g::MatrixGroupElement{N, T}) where {N, T} =
+    isone(word(g)) || matrix_repr(g) == LinearAlgebra.I
 
 function Base.:(==)(m1::M1, m2::M2) where {M1<:MatrixGroupElement, M2<:MatrixGroupElement}
     parent(m1) === parent(m2) || return false
@@ -26,7 +27,9 @@ Base.getindex(sl::MatrixGroupElement, i, j) = matrix_repr(sl)[i,j]
 # Base.iterate(sl::MatrixGroupElement, state) = iterate(sl.elts, state)
 
 function matrix_repr(m::MatrixGroupElement{N, T}) where {N, T}
-    isempty(word(m)) && return StaticArrays.SMatrix{N, N, T}(I)
+    if isone(word(m))
+        return StaticArrays.SMatrix{N, N, T}(LinearAlgebra.I)
+    end
     A = alphabet(parent(m))
     return prod(matrix_repr(A[l]) for l in word(m))
 end

src/matrix_groups/eltary_matrices.jl

@@ -1,6 +1,3 @@
-using Groups
-using StaticArrays
-
 struct ElementaryMatrix{N, T} <: Groups.GSymbol
     i::Int
     j::Int
@@ -24,7 +21,7 @@ Base.inv(e::ElementaryMatrix{N}) where N =
     ElementaryMatrix{N}(e.i, e.j, -e.val)
 
 function matrix_repr(e::ElementaryMatrix{N, T}) where {N, T}
-    m = StaticArrays.MMatrix{N, N, T}(I)
+    m = StaticArrays.MMatrix{N, N, T}(LinearAlgebra.I)
     m[e.i, e.j] = e.val
     x = StaticArrays.SMatrix{N, N}(m)
     return x

src/matrix_groups/eltary_symplectic.jl

@@ -1,6 +1,3 @@
-using Groups
-using StaticArrays
-
 struct ElementarySymplectic{N, T} <: Groups.GSymbol
     symbol::Symbol
     i::Int
@@ -69,7 +66,7 @@ Base.inv(s::ElementarySymplectic{N}) where N =
 function matrix_repr(s::ElementarySymplectic{N, T}) where {N, T}
     @assert iseven(N)
     n = div(N, 2)
-    m = StaticArrays.MMatrix{N, N, T}(I)
+    m = StaticArrays.MMatrix{N, N, T}(LinearAlgebra.I)
     i,j = _ind(s)
     m[i,j] = s.val
     if s.symbol === :A

test/group_constructions.jl

@@ -0,0 +1,43 @@
+@testset "GroupConstructions" begin
+
+    @testset "DirectProduct" begin
+        GH =
+            let G = PermutationGroups.SymmetricGroup(3),
+                H = PermutationGroups.SymmetricGroup(4)
+
+                Groups.Constructions.DirectProduct(G, H)
+            end
+        test_Group_interface(GH)
+        test_GroupElement_interface(rand(GH, 2)...)
+
+        @test collect(GH) isa Array{eltype(GH), 2}
+        @test contains(sprint(print, GH), "Direct product")
+        @test sprint(print, rand(GH)) isa String
+    end
+
+    @testset "DirectPower" begin
+        GGG = Groups.Constructions.DirectPower{3}(
+            PermutationGroups.SymmetricGroup(3),
+        )
+        test_Group_interface(GGG)
+        test_GroupElement_interface(rand(GGG, 2)...)
+
+        @test collect(GGG) isa Array{eltype(GGG), 3}
+        @test contains(sprint(print, GGG), "Direct 3-rd power")
+        @test sprint(print, rand(GGG)) isa String
+    end
+    @testset "WreathProduct" begin
+        W =
+            let G = PermutationGroups.SymmetricGroup(2),
+                P = PermutationGroups.SymmetricGroup(4)
+
+                Groups.Constructions.WreathProduct(G, P)
+            end
+        test_Group_interface(W)
+        test_GroupElement_interface(rand(W, 2)...)
+
+        @test collect(W) isa Array{eltype(W), 2}
+        @test contains(sprint(print, W), "Wreath product")
+        @test sprint(print, rand(W)) isa String
+    end
+end

test/runtests.jl

@@ -33,6 +33,8 @@ include(joinpath(pathof(GroupsCore), "..", "..", "test", "conformance_test.jl"))
     include("AutSigma_41.jl")
     include("AutSigma3.jl")
 
+    include("group_constructions.jl")
+
     # if !haskey(ENV, "CI")
     #    include("benchmarks.jl")
     # end